Reverse Triangle Inequality

What Is the Reverse Triangle Inequality?

The reverse triangle inequality is a mathematical principle stating that for any two real numbers, the absolute value of the difference of their absolute values is less than or equal to the absolute value of their difference. It's a fundamental property of absolute values that helps us establish a 'lower bound' or a minimum possible value for an expression.

While that definition might sound like a mouthful, the formula itself is quite elegant:

Let's break down what this means piece by piece:

• $|a|$ and $|b|$: These are the absolute values of our two numbers, $a$ and $b$. Remember, this means their distance from zero on the number line.
• $|a|-|b|$: This is the difference between their individual distances from zero.
• $||a|-|b||$: This is the absolute value of that difference. We take the absolute value here to ensure the left side is always non-negative, just like the right side. This is the 'difference in their distances from the origin'.
• $|a-b|$: This represents the direct distance between the numbers $a$ and $b$ on the number line.

So, in simple terms, the inequality tells us: The difference in how far two numbers are from zero is always less than or equal to the actual distance between those two numbers. This concept is incredibly useful for setting limits and proving concepts in higher-level mathematics.

First, A Quick Refresher on Absolute Value

Before we dive deeper into the reverse triangle inequality, let's ensure we're solid on the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. We denote the absolute value of a number $x$ with vertical bars: $|x|$.

Because it represents a distance, absolute value is always non-negative (zero or positive).

• The absolute value of a positive number is just the number itself. For example, $|9|=9$.
• The absolute value of a negative number is its positive counterpart. For example, $|-5|=5$.
• The absolute value of zero is zero. $|0|=0$.

A crucial idea for our topic is that the expression $|a-b|$ represents the distance between $a$ and $b$ on the number line. For instance, the distance between $10$ and $4$ is $|10-4|=|6|=6$. The distance between $10$ and $-4$ is $|10-(-4)|=|10+4|=|14|=14$. Understanding absolute value as a measure of distance is the key to unlocking both the standard and reverse triangle inequalities.

How Does It Relate to the 'Standard' Triangle Inequality?

The reverse triangle inequality has a more famous sibling: the standard Triangle Inequality. Understanding how they relate to each other provides a more complete picture. The standard triangle inequality states:

This formula tells us that the absolute value of a sum is always less than or equal to the sum of the absolute values. Think of it this way: the shortest path from the origin to the point $a+b$ is a straight line. Taking a detour through another point (going from the origin to $a$, then from $a$ to $a+b$) is always longer or the same length. It provides an upper bound—the maximum possible value of $|a+b|$.

The reverse triangle inequality, on the other hand, provides a lower bound—the minimum possible value of $|a-b|$. They are two sides of the same coin, helping us bracket an expression from above and below.

Here is a table to help you keep them straight:

How Can We Prove the Reverse Triangle Inequality?

In mathematics, we don't just accept rules; we prove them! The good news is that we can prove the reverse triangle inequality using the standard triangle inequality, which we'll assume is true. The logic is a clever application of substitution and algebra.

Let's walk through the steps. Our goal is to prove $||a|-|b||\le |a-b|$.

1. First, let's consider the number $a$. We can use a simple trick to rewrite it in terms of $a-b$ and $b$: $a=(a-b)+b$.
2. Now, take the absolute value of both sides: $|a|=|(a-b)+b|$.
3. Apply the standard triangle inequality ($|x+y|\le |x|+|y|$) to the right side, where $x=a-b$ and $y=b$:
   $|a|\le |a-b|+|b|$
4. Subtract $|b|$ from both sides of the inequality. This gives us our first result:
   $|a|-|b|\le |a-b|$
5. Next, we do the same thing, but starting with $b$. Rewrite $b$ as $b=(b-a)+a$.
6. Take the absolute value of both sides: $|b|=|(b-a)+a|$.
7. Apply the standard triangle inequality to the right side: $|b|\le |b-a|+|a|$.
8. Remember that the distance from $b$ to $a$ is the same as the distance from $a$ to $b$. Mathematically, $|b-a|=|a-b|$. So we can substitute that in:
   $|b|\le |a-b|+|a|$
9. Subtract $|a|$ from both sides: $|b|-|a|\le |a-b|$.
10. Now we have two key statements:
    (1) $|a|-|b|\le |a-b|$
    (2) $|b|-|a|\le |a-b|$
    Notice that $|b|-|a|$ is the same as $-(|a|-|b|)$. So our second statement can be written as $-(|a|-|b|)\le |a-b|$.
11. If a value $X$ satisfies both $X\le C$ and $-X\le C$, that is the definition of $|X|\le C$. In our case, $X=|a|-|b|$ and $C=|a-b|$. Therefore, we can combine our two results into a single statement:

And with that, the proof is complete!

Solving Problems with the Reverse Triangle Inequality: Worked Examples

The best way to understand the reverse triangle inequality is to see it in action. Let's test it with different kinds of numbers to verify that it always holds true.

Example 3

Let $a=-15$ and $b=-6$. (Two negative numbers)

We need to check if $||a|-|b||\le |a-b|$ is true.

1. Calculate the LHS: $||a|-|b||$
   Find the absolute values: $|-15|=15$ and $|-6|=6$.
   Substitute them: $|15-6|=|9|=9$.
   The LHS is $9$.
2. Calculate the RHS: $|a-b|$
   Substitute $a$ and $b$: $|-15-(-6)|$.
   $|-15+6|=|-9|=9$.
   The RHS is $9$.
3. Compare the results: Is $9\le 9$? Yes, it is. The inequality holds, and this is another case of equality.

What Does This Look Like on a Number Line?

Visualizing the reverse triangle inequality on a number line can make the concept much more intuitive. Let's use the values from Example 2: $a=5$ and $b=-8$.

The two sides of the inequality $||a|-|b||\le |a-b|$ represent two different kinds of distances.

The Right-Hand Side: $|a-b|$

This term, $|5-(-8)|=13$, represents the direct distance between the points $5$ and $-8$ on the number line. If you were to count the units between $-8$ and $5$, you would count $13$ spaces. It's the most straightforward distance calculation.

The Left-Hand Side: $||a|-|b||$

This term represents something different. It's the difference of their individual distances from the origin (zero).

• The distance of $a=5$ from zero is $|5|=5$.
• The distance of $b=-8$ from zero is $|-8|=8$.
• The expression $||a|-|b||$ becomes $|5-8|=|-3|=3$.

So, the inequality for this example is $3\le 13$, which is true. The visualization shows us why it's true. The direct distance between the two points ($13$) is much larger than the difference in their distances from the origin ($3$).

When does equality happen, like in Examples 1 and 3? Equality $||a|-|b||=|a-b|$ occurs when $a$ and $b$ are on the same side of zero (or one is zero). In this case, the direct distance between them is exactly the same as the difference of their distances from the origin.

Common Mistakes and Pitfalls

When working with the reverse triangle inequality, a few common errors can trip students up. Being aware of them is the first step to avoiding them.

• Forgetting the Outer Absolute Value: A very common mistake is to calculate $|a|-|b|$ instead of $||a|-|b||$. The result of the left-hand side must be non-negative. Forgetting the outer bars can lead to a negative number, which would make the inequality seem incorrect. For $a=5,b=-8$, $|5|-|-8|=5-8=-3$, but the correct LHS is $||5|-|-8||=|5-8|=|-3|=3$.
• Confusing It with the Standard Inequality: It's easy to mix up the formulas. Remember, the reverse inequality involves subtraction on both sides ($||a|-|b||$ and $|a-b|$) and provides a lower bound. The standard inequality involves addition ($|a+b|$ and $|a|+|b|$) and provides an upper bound.
• Sign Errors in Subtraction: When calculating $|a-b|$, be extremely careful with negative numbers. A common error is writing $|5--8|$ as $|5-8|$. Always simplify the expression inside the absolute value bars first: $5-(-8)=5+8=13$.
• Assuming Strict Inequality: The rule is "less than or equal to" ($\le$). As we saw in two of our examples, equality is possible and occurs frequently. Don't assume the left side must always be strictly smaller than the right side.

Quick Summary and Key Formulas

This lesson covered a lot of ground. Here are the most important takeaways and formulas to remember.

• The Reverse Triangle Inequality provides a lower bound, or a minimum value, for the distance between two numbers.
• It states that the difference in the distances of two numbers from the origin is always less than or equal to the direct distance between the two numbers.
• It is derived from the standard Triangle Inequality.
• Equality holds when the numbers $a$ and $b$ have the same sign or one of them is zero.

Keep these two fundamental formulas handy as a reference: